2.3.1. When enough is enough

So far we have seen only derivations whose gaps all close, derivations which show that arguments are valid. But not all arguments are valid, so there ought to be derivations whose gaps do not all close. And if there is no point at which the gaps of a derivation all close, we will eventually have to give up work on it even though it still has open gaps. So we should ask if there can be a reason for giving up work and what, if anything, we can conclude about an argument if we have grounds for giving up work on a derivation for it.

The short answer to the first of these two questions is that we must give up on a derivation when we run out of rules to apply, either to develop a gap or close it. Here’s a simple example of a derivation for which that has happened.

The gap that is marked with the empty circle ○ has C as its goal, and we currently have no rule to plan for such a goal. There are conjunctions among the available resources of the gap; but they were exploited in the course of developing this gap, so they are no longer active. Also, none of the rules for closing gaps apply here: not QED because the goal is not one of available resources, not EFQ because ⊥ is not a resource, and not ENV because the goal is not ⊤. In short, no rule of any of the three sorts can be applied at this point.

The resources added by exploiting A ∧ ⊤ at stage 2 were never used later (hence there are no line numbers to their right). As a result, this exploitation could have been postponed the end. But, once it is done, there is no more to do. However, although it is easy to see that it will not lead the gap to close, the resource A ∧ ⊤ would have to be exploited before we could claim to have ended because no more that can be done. Until it is exploited, there is a way of developing the derivation further. One thing that needn’t have been done is closing the first gap at stage 4. As long as one gap has reached a point where no more can be done to close it, there is reason to stop because all gaps must close before the derivation is complete.

We will describe an open gap to which no more rules apply as a dead-end gap. (Although the qualification dead-end will be reserved for open gaps—indeed, a gap that has been closed is in one sense no longer a gap—we will often speak somewhat redundantly of dead-end open gaps.) In these terms, we can say that we are forced to abandon a derivation when every open gap has reached a dead end although we may abandon a derivation as soon as one open gap has reached a dead end. As in the example above, we will use the empty circle to mark open gaps that have reached a dead end and are thus permanently open. And, also as is done in that example, to the right of this sign, we will use the sign ⊭ (negated double right turnstile) to say that, with respect to the analysis of them displayed in the derivation, the active resources do not entail the goal. (The reason for qualifying this by reference to the displayed analysis will be discussed in 2.3.8.)

The way the gaps have developed in this derivation is shown in the following skeleton tree with the full argument tree below it:

 

 

 

 

 

 

 

(A ∧ ⊤) ∧ B

---

B ∧ C

A ∧ ⊤
B

---

B ∧ C

B
A
⊤

---

B ∧ C

B
A
⊤

---

B

B
A
⊤

---

C

B
A
⊤

---

C

The gap that remains open at the end had reached a dead end at stage 3, but it is shown to continue at the next stage because it remains open as the derivation develops elsewhere. Although work may be stopped as soon as a dead end is reached, there is nothing wrong with continuing as long as there are rules to be applied to other gaps, and that will often be done in examples. In general we will not assume that work on a derivation stops as soon as there is a dead-end gap, so to say that gap has reached a dead end is not to say that the gap does not continue at later stages; it is to say rather that it cannot be developed further.

We will now turn to considering the significance of dead end gaps. We will look first at what reaching a dead end tells us about the proximate argument of the gap that has stopped developing and then consider the connection between the validity of the ultimate argument of a derivation and the existence of dead-end gaps. In terms of argument trees, this means we will look first at the tips of unclosed branches and then ask about the connection between the tips of branches and the root of the tree.